3.7.32 \(\int \frac {\log ^n(\log (x))}{x} \, dx\) [632]

Optimal. Leaf size=24 \[ \Gamma (1+n,-\log (\log (x))) (-\log (\log (x)))^{-n} \log ^n(\log (x)) \]

[Out]

GAMMA(1+n,-ln(ln(x)))*ln(ln(x))^n/((-ln(ln(x)))^n)

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2336, 2212} \begin {gather*} (-\log (\log (x)))^{-n} \log ^n(\log (x)) \text {Gamma}(n+1,-\log (\log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[Log[x]]^n/x,x]

[Out]

(Gamma[1 + n, -Log[Log[x]]]*Log[Log[x]]^n)/(-Log[Log[x]])^n

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2336

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rubi steps

\begin {align*} \int \frac {\log ^n(\log (x))}{x} \, dx &=\text {Subst}\left (\int \log ^n(x) \, dx,x,\log (x)\right )\\ &=\text {Subst}\left (\int e^x x^n \, dx,x,\log (\log (x))\right )\\ &=\Gamma (1+n,-\log (\log (x))) (-\log (\log (x)))^{-n} \log ^n(\log (x))\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} \Gamma (1+n,-\log (\log (x))) (-\log (\log (x)))^{-n} \log ^n(\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[Log[x]]^n/x,x]

[Out]

(Gamma[1 + n, -Log[Log[x]]]*Log[Log[x]]^n)/(-Log[Log[x]])^n

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (\ln \left (x \right )\right )^{n}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(ln(x))^n/x,x)

[Out]

int(ln(ln(x))^n/x,x)

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Maxima [A]
time = 0.47, size = 29, normalized size = 1.21 \begin {gather*} -\left (-\log \left (\log \left (x\right )\right )\right )^{-n - 1} \log \left (\log \left (x\right )\right )^{n + 1} \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^n/x,x, algorithm="maxima")

[Out]

-(-log(log(x)))^(-n - 1)*log(log(x))^(n + 1)*gamma(n + 1, -log(log(x)))

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Fricas [A]
time = 0.25, size = 14, normalized size = 0.58 \begin {gather*} \cos \left (\pi n\right ) \Gamma \left (n + 1, -\log \left (\log \left (x\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^n/x,x, algorithm="fricas")

[Out]

cos(pi*n)*gamma(n + 1, -log(log(x)))

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Sympy [A]
time = 0.89, size = 24, normalized size = 1.00 \begin {gather*} \left (- \log {\left (\log {\left (x \right )} \right )}\right )^{- n} \log {\left (\log {\left (x \right )} \right )}^{n} \Gamma \left (n + 1, - \log {\left (\log {\left (x \right )} \right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(ln(x))**n/x,x)

[Out]

log(log(x))**n*uppergamma(n + 1, -log(log(x)))/(-log(log(x)))**n

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(log(x))^n/x,x, algorithm="giac")

[Out]

integrate(log(log(x))^n/x, x)

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Mupad [B]
time = 0.45, size = 24, normalized size = 1.00 \begin {gather*} \frac {{\ln \left (\ln \left (x\right )\right )}^n\,\Gamma \left (n+1,-\ln \left (\ln \left (x\right )\right )\right )}{{\left (-\ln \left (\ln \left (x\right )\right )\right )}^n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(log(x))^n/x,x)

[Out]

(log(log(x))^n*igamma(n + 1, -log(log(x))))/(-log(log(x)))^n

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